Hybrid orbital approach

Molecular orbital theory is more complex than the hybrid orbital approach, but the foundations of the model are readily accessible. Though complex, molecular orbital theory opens the door to many fascinating aspects of modem chemistry. In this section, we introduce the molecular orbital approach through diatomic molecules. 

The hybridized orbital approach is a simplified way of predicting the geometry of a molecule by mixing the valence orbitals of its atoms. For example, methane (CH ) is composed of a carbon atom with an electron configuration of Is 2s 2p. The hydrogen atom has an electron configuration of Is. The geometry of the methane... 

There is a close relationship between the VSEPR theory discussed in Section 3.9 and the hybrid orbital approach, with steric numbers of 2, 3, and 4 corresponding to sp, sp, and sp hybridization, respectively. The method can be extended to more complex structures (fsp hybridization (see Sec. 8.7), which gives six equivalent hybrid orbitals pointing toward the vertices of a regular octahedron, is applicable to molecules with steric number 6. Both theories are based on minimizing the energy by reducing electron-electron repulsion. 

Describe the bonding in important functional groups using the hybrid orbitals approach (Section 7.6, Problems 27-30). 

Whether the bonding in lithium alkyls is predominantly ionic or covalent is still a matter for debate. Assuming a covalent model, use a hybrid orbital approach to suggest a bonding scheme for (MeLi). Comment on the bonding picture you have described. 

Conceptually the hybrid orbital approach is to be preferred over the link atom method as no unphysical atoms are introduced into the system. Undoubtedly the HO method will become more popular but, as we shall see in section 4, link atoms have been the most widely used when studying macromolecular systems, in large part because they are simpler to implement. 

When two sp2-hybridized carbons approach each other, they form a cr bond by sp2-sp2 head-on overlap. At the same time, the unhybridized p orbitals approach with the correct geometry for sideways overlap, leading to the formation of what is called a pi (ir) bond. The combination of an >p2-sp2 a bond and a 2p-2p 77 bond results iii the sharing of four electrons and the formation of a carbon-carbon double bond (Figure 1.14). Note that the electrons in then-bond occupy the region centered between nuclei, while the electrons in the 77 bond occupy regions on either side of a line drawn between nuclei. 

When two sp-hybridized carbon atoms approach each other, sp hybrid orbitals on each carbon overlap head-on to form a strong sp-sp a bond. In addition, the pz orbitals from each carbon form a pz-pz it bond by sideways overlap and the py orbitals overlap similarly to form a py-py tt bond. The net effect is the sharing of six electrons and formation of a carbon-carbon triple bond. The two remaining sp hybrid orbitals each form a bond with hydrogen to complete the acetylene molecule (Figure 1.16). 

We said in Section 1.5 that chemists use two models for describing covalent bonds valence bond theory and molecular orbital theory. Having now seen the valence bond approach, which uses hybrid atomic orbitals to account for geometry and assumes the overlap of atomic orbitals to account for electron sharing, let s look briefly at the molecular orbital approach to bonding. We ll return to the topic in Chapters 14 and 15 for a more in-depth discussion. 

You might recall from Section 1.9 that a carbon-carbon triple bond results from the interaction of two sp-hybridized carbon atoms. The two sp hybrid orbitals of carbon lie at an angle of 180° to each other along an axis perpendicular to the axes of the two unhybridized 2py and 2pz orbitals. When two sp-hybridized carbons approach each other, one sp-sp a bond and two p-p -rr bonds are... 

The formation of the BeF2 molecule can be explained by assuming that, as two fluorine atoms approach Be, the atomic orbitals of the beryllium atom undergo a significant change. Specifically, the 2s orbital is mixed or hybridized with a 2p orbital to form two new sp hybrid orbitals. (Figure 7.12). 

When combining QM with MM methods, the partitioning of the system will often intersect a chemical bond. This bond is usually chosen to be a carbon-carbon single bond (whenever possible) and three major coupling methods have been developed, which are referred to as the link-atom [54] , pseudo-atom/bond [55] and hybrid-orbital [56] approach, respectively. In the link atom approach the open valency at the border is capped by a hydrogen atom, and most DFTB QM/MM implementations are based on this simple scheme [49, 50] or related variations [57], Recently,... 

Pauling further extended the sp"dm hybridization approach to the d-block compounds.3 By varying the relative importance of p and d orbitals, Pauling was able to construct hybrid orbitals that rationalized the geometries and magnetic properties of many transition-metal coordination complexes. For example, the square-planar... 

We now use a Pauling-like approach to show how hybrid orbitals for a variety of combinations of s, p, and d orbitals may be formulated.10 We assume that the radial dependences of the s, p, d orbitals are similar so that they can be neglected. The angular parts of the orbital wavefunctions are given by the following expressions (in the usual spherical coordinates 9, ) ... 

To circumvent problems associated with the link atoms different approaches have been developed in which localized orbitals are added to model the bond between the QM and MM regions. Warshel and Levitt [17] were the first to suggest the use of localized orbitals in QM/MM studies. In the local self-consistent field (LSCF) method the QM/MM frontier bond is described with a strictly localized orbital, also called a frozen orbital [43]. These frozen orbitals are parameterized by use of small model molecules and are kept constant in the SCF calculation. The frozen orbitals, and the localized orbital methods in general, must be parameterized for each quantum mechanical model (i.e. energy-calculation method and basis set) to achieve reliable treatment of the boundary [34]. This restriction is partly circumvented in the generalized hybrid orbital (GHO) method [44], In this method, which is an extension of the LSCF method, the boundary MM atom is described by four hybrid orbitals. The three hybrid orbitals that would be attached to other MM atoms are fixed. The remaining hybrid orbital, which represents the bond to a QM atom, participates in the SCF calculation of the QM part. In contrast with LSCF approach the added flexibility of the optimized hybrid orbital means that no specific parameterization of this orbital is needed for each new system. 

Separation of covalently bonded atoms into QM and MM regions introduces an unsatisfied valence in the QM region this can be satisfied by several different methods. In the frozen-orbital approach a strictly localized hybrid sp2 bond orbital containing a single electron is used at the QM/MM junction [29]. Fro-... 

An important point to be stressed before we proceed further with our analysis concerns the formal correspondence, of hybrid orbitals and delocalized group orbitals. For example, consider the model system HN=NH which can exist in a cis or a tram geometry. We can understand whether sigma interactions favor one or the other geometry be means of one of the following two approaches ... 

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